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Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy

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Abstract.

We consider the Cauchy problem for a general one-dimensional n×n hyperbolic symmetrizable system of balance laws. It is well known that, in many physical examples, for instance for the isentropic Euler system with damping, the dissipation due to the source term may prevent the shock formation, at least for smooth and small initial data. Our main goal is to find a set of general and realistic sufficient conditions to guarantee the global existence of smooth solutions, and possibly to investigate their asymptotic behavior. For systems which are entropy dissipative, a quite natural generalization of the Kawashima condition for hyperbolic-parabolic systems can be given. In this paper, we first propose a general framework for this kind of problem, by using the so-called entropy variables. Then we go on to prove some general statements about the global existence of smooth solutions, under different sets of conditions. In particular, the present approach is suitable for dealing with most of the physical examples of systems with a relaxation extension. Our main tools will be some refined energy estimates and the use of a suitable version of the Kawashima condition.

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References

  1. Aregba-Driollet, D., Natalini, R.: Convergence of relaxation schemes for conservation laws. Appl. Anal. 61, 163–193 (1996)

    MATH  Google Scholar 

  2. Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000) (electronic)

    MathSciNet  MATH  Google Scholar 

  3. Bianchini, S.: A Glimm type functional for a special Jin-Xin relaxation model. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 19–42 (2001)

    Article  MATH  Google Scholar 

  4. Boillat, G.: Sur l'Existence et la Recherche d'Équations de Conservation Supplémentaires pour les Systèmes Hyperboliques. C.R. Acad. Sci. Paris A 278, 909–912 (1974)

    MATH  Google Scholar 

  5. Boillat, G., Ruggeri, T.: Hyperbolic principal subsystems: entropy convexity and subcharacteristic conditions. Arch. Rational Mech. Anal. 137, 305–320 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boillat, G., Ruggeri, T.: On the shock structure problem for hyperbolic system of balance laws and convex entropy. Contin. Mech. Thermodyn. 10, 285–292 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation law. J. Statist. Phys. 113–170 (1999)

  8. Chen, G.-Q., Levermore, C.D., Liu, T.-P.: Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Dafermos, C.M.: A system of hyperbolic conservation laws with frictional damping. Z. Angew. Math. Phys. 46 (1995), no. Special Issue, S294–S307, Theoretical, experimental, and numerical contributions to the mechanics of fluids and solids

  10. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Springer-Verlag, Berlin, xvi+443 pp., 2000

  11. Friedrichs, K.O., Lax, P.D.: Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. U.S.A. 68, 1686–1688 (1971)

    MATH  Google Scholar 

  12. Giovangigli, V., Massot, M.: Asymptotic Stability of Equilibrium States for Multicomponent Reacting Flows. Math. Models Meth. Appl. Sci. 2, 251–297 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Giovangigli, V., Massot, M.: The Local Cauchy Problem for Multicomponent Reactive Flows in Full Vibrational Nonequilibrium. Math. Meth. Appl. Sci. 1415–1439 (1998)

  14. Godunov, S.K.: An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR 139, 521–523 (1961)

    MATH  Google Scholar 

  15. Hanouzet, B., Huynh, Ph.: Approximation par relaxation d'un système de Maxwell non linéaire. C. R. Acad. Sci. Paris Sér. I Math. 330, 193–198 (2000)

    Article  MATH  Google Scholar 

  16. Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Physics 143, 599–605 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Hsiao, L., Pan, R.: Zero relaxation limit to centered rarefaction waves for a rate-type viscoelastic system. J. Differential Equations 157, 20–40 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hsiao, L., Serre, D.: Global existence of solutions for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal. 27, 70–77 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Huynh, Ph.: Étude théorique et numérique de modèles de Kerr. Ph.D. thesis, Université Bordeaux I, 1999

  20. Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Judkins, J.B., Ziolkowski, R-W.: Full wave vector Maxwell equation modeling of the self–focusing of ultrashort optical pulses in a non linear Kerr medium exhibiting a finite response time. J. Opt. Soc. Am. B 45, 375–391 (1997)

    Google Scholar 

  22. Kawashima, S.: Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications. Proc. Roy. Soc. Edinburgh Sect. A 106, 169–194 (1987)

    MathSciNet  MATH  Google Scholar 

  23. Li, T.T.: Global classical solutions for quasilinear hyperbolic systems. Masson, Paris; John Wiley & Sons, Ltd., Chichester, x+315 pp., 1994

  24. Liu, T.-P.: Hyperbolic conservation laws with relaxation. Commun. Math. Phys. 108, 153–175 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Li, H., Pan, R.: Zero relaxation limit for piecewise smooth solutions to a rate-type viscoelastic system in the presence of shocks. J. Math. Anal. Appl. 252, 298–324 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Majda, A.: Compressible fluid flow and systems of conservation laws in several space dimensions. Appl. Math. Sci., vol. 53, Springer–Verlag, 1978

  27. Massot, M.: Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure. Discrete and Continuous Dynamical Systems – Series B 2, 433–456 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Müller, I., Ruggeri, T.: Rational extended thermodynamics. Second ed., Springer-Verlag, New York, 1998, With supplementary chapters by H. Struchtrup and Wolf Weiss

  29. Natalini, R.: Convergence to equilibrium for the relaxation approximation of conservation laws. Commun. Pure Appl. Math. 49, 795–823 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  30. Natalini, R.: A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Differential Equations 148, 292–317 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Natalini, R.: Recent results on hyperbolic relaxation problems. Analysis of systems of conservation laws (Aachen, 1997), Chapman & Hall/CRC, Boca Raton, FL, pp. 128–198, 1999

  32. Nishida, T.: Nonlinear hyperbolic equations and related topics in fluid dynamics, Département de Mathématique, Université de Paris-Sud, Orsay, Publications Mathématiques d'Orsay, No. 78–02, 1978

  33. Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. To appear in Quart Appl Math.

  34. Serre, D.: Relaxations semi-linéaire et cinétique des systèmes de lois de conservation. Ann. Inst. H. Poincaré Anal. Non Linéaire 17, 169–192 (2000)

    Article  MATH  Google Scholar 

  35. Shizuta, Y., Kawashima, S.: Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math. J. 14, 249–275 (1985)

    Google Scholar 

  36. Sideris, T., Thomases, B., Wang, D.: Decay and singularities of solutions of the three-dimensional Euler equations with damping. To appear in Comm. PDE.

  37. Suliciu, I.: On modelling phase transitions by means of rate-type constitutive equations. Shock wave structure. Internat. J. Engng. Sci. 28, 829–841 (1990)

    Article  MATH  Google Scholar 

  38. Tzavaras, A.E.: Materials with internal variables and relaxation to conservation laws. Arch. Rational. Mech. Anal. 146, 129–155 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. Whitham, G.B.: Linear and nonlinear waves. A Wiley & Sons, New York, 1974

  40. Yong, W.-A.: Basic aspects of hyperbolic relaxation systems. Advances in the theory of shock waves, Birkhäuser Boston, Boston, MA, pp. 259–305, 2001

  41. Yong, W.-A.: An entropy condition and global existence result for hyperbolic balance laws. Preprint, December 2002

  42. Zeng, Y.: Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation. Arch. Rational. Mech. Anal. 150, 225–279 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathématiques Appliquées de Bordeaux, Université Bordeaux 1, 351 cours de la Libération, F-33405, Talence, France

    B. Hanouzet

  2. Istituto per le Applicazioni del Calcolo ``Mauro Picone'', Consiglio Nazionale delle Ricerche, Viale del Policlinico, 137, I-00161, Roma, Italy

    R. Natalini

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  1. B. Hanouzet
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  2. R. Natalini
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Correspondence to R. Natalini.

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Communicated by C.M. Dafermos

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Hanouzet, B., Natalini, R. Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy. Arch. Rational Mech. Anal. 169, 89–117 (2003). https://doi.org/10.1007/s00205-003-0257-6

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